Distinguished positive regular representations

Let G be a tamely ramified reductive p-adic group. We study distinction of a class of irreducible admissible representations of G by the group of fixed points H of an involution of G. The representations correspond to G-conjugacy classes of pairs (T,ϕ), where T is a tamely ramified maximal torus of G and ϕ is a quasicharacter of T whose restriction to the maximal pro-p-subgroup satisfies a regularity condition. Under mild restrictions on the residual characteristic of F, we derive necessary conditions for H-distinction of a representation corresponding to (T,ϕ), expressed in terms of properties of T and ϕ relative to the involution. We prove that if an H-distinguished representation arises from a pair (T,ϕ) such that T is stable under the involution and compact modulo (T∩H)Z (here, Z is the centre of G), then the representation is H-relatively supercuspidal.